Course Syllabus

Quick links:

• Homework assignments
• Lecture notes

Content: This course provides a rigorous introduction to abstract algebra, including group theory and linear algebra.  The formal prerequisites for Math 55 are minimal, but this class does require a commitment to a demanding course, strong interest in mathematics, and some familiarity with proofs and abstract reasoning. 

Important: Due to the course's place in the first-year undergraduate mathematics experience at Harvard and its role in helping our future math concentrators get to know each other and form a community, Math 55 is only open to first-year Harvard College students. 

Lectures: will be held Mondays, Wednesdays, Fridays from 10:30 to 11:45, starting Wednesday September 4, in Science Center 507. Students are expected to attend all lectures. Handwritten lecture notes will be provided.

Course staff: 

• Prof. Denis Auroux (auroux@math.harvard.edu)  (office hours MW 12:30-1:30 in SC 539; also Fri 12:30-1:30 on Sept 6, 13, 20).
• Course Assistants: Jinho Park, Cyrus Hamlin, Rohan Nambiar, Sophie Zhu, Stella Li, Enrico Yao-Bate, Jacob Paltrowitz, Madeleine de Belloy, Halyna Bowey, Alvan Arulandu, Stephen Yang

CA office hours: (starting Friday Sept 6):

• Sundays: Cyrus + Halyna, 7:30-8:30pm, Math Lounge
• Mondays: Rohan + Stella, 4-5:30pm, Math Lounge
• Tuesdays: Jinho + Sophie, 7:30-9pm, Math Lounge
• Wednesdays: Enrico + Stephen, 8-10pm, Math Lounge
• Thursdays: Alvan, 3-4pm, Lowell Dining Hall
• Fridays: Jacob + Madeleine, 1-3pm, Science Center 530

The CAs will also hold various one-time discussion sections in order to go over specific material or skill sets (proof writing, LaTeX, review sessions, etc.).  Even if you don't need help with specific problems on the current assignment, attending some office hours or sections is recommended -- it gives you an opportunity to review and ask general questions about the course material (or related topics).  It's also a chance to get to know the CAs and your fellow students in smaller groups.

Textbooks: there are two required texts, Axler's Linear Algebra Done Right and Artin's Algebra.  (We will aim to cover most of Axler, and most of chapters 2-10 in Artin).
Other books that may (or may not) be helpful at various points in the course include Dummit and Foote's Abstract Algebra, Halmos' Naive Set Theory , Halmos' Finite-Dimensional Vector Spaces , Fulton and Harris' Representation Theory: A First Course , and Serre's  Linear Representations of Finite Groups .   Electronic versions of all of these books except for Artin and Dummit & Foote are freely available to Harvard students from the Springer-Verlag website (by clicking on the links above).

Homework will be assigned weekly, and is due on Wednesday each week.  Assignments will be posted on this site, and should be submitted electronically via this website.
Doing the homework in a timely fashion is essential to learning the material properly; given the pace, it is extremely hard to catch up if you fall behind in this class. 
Extensions will be granted for illness or other serious circumstances (having overcommitted yourself does not count), but should be requested ahead of the deadline; outside of these cases, late submissions will incur a penalty to be computed at the end of the semester once your cumulative lateness across all assignments exceeds 96 hours (4 days).

Exams: there will be a midterm (take-home, will be posted Sept 27, due Oct 2) and a take-home final exam (will be posted Dec 5, due Dec 11).

Course grades will be based on your homework (65%), the midterm (10%), and the final exam (25%).
One homework score will be dropped for everyone, so you may miss one assignment without penalty, but you are still responsible for working through the material.

Community:  One of the best features of Math 55 is the sense of community that most students get out of it.  Getting to know the CAs and your fellow students early in the semester, and forming study groups,  is an important part of the experience -- they'll be your support network when the math gets rough. Drop by office hours to introduce yourself and meet others (even if you don't have any specific questions). Participate in the Slack discussions. And please remember: it is up to you to make this community inclusive, welcoming, and supportive of all of its members -- and of all the other people around you, including those who aren't in Math 55.

Academic integrity policy: You are encouraged to discuss and collaborate with each other on the homework assignments. However,  make sure that you can work through the problems yourself, and write up your answers on your own. This is not only a matter of academic integrity, but also crucial for properly learning the material and the problem-solving skills that this course aims to cover.  For exams, collaboration or consultation of sources other than those explicitly permitted is not allowed.

Homework assignments

Direct links to the homework assignments will be posted here as PDF files. Go to "Assignments" to see the LaTeX source, and to submit your solutions. Note: HW3 and beyond will be submitted via Gradescope (submission link on each assignment page)

• Homework 0 Download Homework 0 (warm-up, not due)
• Homework 1 Download Homework 1 (due Wed Sep 11)
• Homework 2 Download Homework 2 (due Wed Sep 18)
• Homework 3 Download Homework 3 (due Wed Sep 25)
• Homework 4 Download Homework 4 (due Wed Oct 2, along with the midterm)
• Homework 5 Download Homework 5 (due Wed Oct 9)
• Homework 6 Download Homework 6 (due Wed Oct 16)
• Homework 7 Download Homework 7 (due Wed Oct 23)
• Homework 8 Download Homework 8 (due Wed Oct 30)
• Homework 9 Download Homework 9 (due Wed Nov 6)
• Homework 10 Download Homework 10 (due Wed Nov 13)
• Homework 11 Download Homework 11 (due Wed Nov 20)
• Homework 12 Download Homework 12 (due Wed Dec 4)

List of Topics and Lecture Notes

Direct links to the lecture notes will be posted here as PDF files.

Here is a tentative list of topics / dates; we may deviate from these.

• Lecture 1 Download Lecture 1 (Wed Sep 4): Course logistics; groups, examples
• Lecture 2 Download Lecture 2 (Fri Sep 6): Set theory; more examples; subgroups; homomorphisms
• Lecture 3 Download Lecture 3 (Mon Sep 9): Subgroups of Z; cyclic groups; equivalence relations; cosets
• Lecture 4 Download Lecture 4 (Wed Sep 11): Cosets, normal subgroups, quotient groups; exact sequences
• Lecture 5 Download Lecture 5 (Fri Sep 13): Exact sequences; the symmetric group; center, commutators; free groups; rings and fields
• Lecture 6 Download Lecture 6 (Mon Sep 16, online): Rings and fields; vector spaces; independence, span, basis, dimension
• Lecture 7 Download Lecture 7 (Wed Sep 18, online): Bases, dimension; linear maps and matrices; direct sums; rank, dimension formula
• Lecture 8 Download Lecture 8 (Fri Sep 20): Rank and dimension formula; change of basis; quotient spaces; dual spaces
• Lecture 9 Download Lecture 9 (Mon Sep 23): Quotients, duals and transpose; linear operators, invariant subspaces
• Lecture 10 Download Lecture 10 (Wed Sep 25): Linear operators; invariant subspaces, eigenvectors, eigenvalues; triangular matrices
• Lecture 11 Download Lecture 11 (Fri Sep 27): Triangular matrices and eigenvalues; generalized kernel, generalized eigenspaces; nilpotent operators; Jordan normal form
• Lecture 12 Download Lecture 12 (Mon Sep 30): Jordan normal form; characteristic and minimal polynomials; real operators; categories and functors (handout Download handout)
• Lecture 13 Download Lecture 13 (Wed Oct 2): Categories and functors (handout Download handout); bilinear forms; orthogonality; inner products
• Lecture 14 Download Lecture 14 (Fri Oct 4): Orthogonality; inner products; orthonormal bases; orthogonal operators
• Lecture 15 Download Lecture 15 (Mon Oct 7): Orthogonal and self-adjoint operators; spectral theorem for real operators; Hermitian inner products
• Lecture 16 Download Lecture 16 (Wed Oct 9): Hermitian inner products; the complex spectral theorem; classifying bilinear forms
• Lecture 17 Download Lecture 17 (Fri Oct 11): Skew-symmetric bilinear forms; tensor product: definition and properties; trace (handout) Download (handout)
• Lecture 18 Download Lecture 18 (Wed Oct 16): Tensor algebra; symmetric and exterior algebras; volume and determinant (handout) Download (handout)
• Lecture 19 Download Lecture 19 (Fri Oct 18): Volume and determinant; modules over rings
• Lecture 20 Download Lecture 20 (Mon Oct 21): Classification of finitely generated abelian groups; group actions
• Lecture 21 Download Lecture 21 (Wed Oct 23): Group actions, orbits and stabilizers; Burnside's lemma; conjugacy classes
• Lecture 22 Download Lecture 22 (Fri Oct 25): Finite subgroups of SO(3) and regular polyhedra
• Lecture 23 Download Lecture 23 (Mon Oct 28): Conjugacy classes; groups of order p²; conjugacy in the symmetric group
• Lecture 24 Download Lecture 24 (Wed Oct 30): Conjugacy in S_n and A_n; statement of Sylow theorems
• Lecture 25 Download Lecture 25 (Fri Nov 1): Statement of Sylow theorems; examples, semi-direct products
• Lecture 26 Download Lecture 26 (Mon Nov 4): Proof of Sylow theorems; groups of order 12
• Lecture 27 Download Lecture 27 (Wed Nov 6): Groups of order 12; group presentations, Cayley graph, normal forms
• Lecture 28 Download Lecture 28 (Fri Nov 8): Presentations and normal forms: S_n, braid group, SL(2,Z)
• Lecture 29 Download Lecture 29 (Mon Nov 11): Representations; sub-representations; irreducibility
• Lecture 30 Download Lecture 30 (Wed Nov 13): Irreducibility; Schur's lemma; representations of S_3
• Lecture 31 Download Lecture 31 (Fri Nov 15): Symmetric polynomials; the character of a representation
• Lecture 32 Download Lecture 32 (Mon Nov 18): Characters, orthogonality, and consequences; characters of S_4 and A_4
• Lecture 33 Download Lecture 33 (Wed Nov 20): Irreducible characters form a basis; the representation ring
• Lecture 34 Download Lecture 34 (Fri Nov 22): Irreducible characters of S_5 and A_5; restriction and induction
• Lecture 35 Download Lecture 35 (Mon Nov 25): Restriction and induction, Frobenius reciprocity
• Lecture 36 Download Lecture 36 (Mon Dec 2): Real and quaternionic representations; wrap-up and final info
• Lecture 37 Download Lecture 37 (Wed Dec 4): Wrap-up, final info, and review session
• Review part 1 Download part 1 (linear algebra) and part 2 Download part 2 (group theory and representations)